Invertible functions

Definition

An invertible function is a function that has an inverse, meaning there exists another function that 'undoes' the effect of the original function. For a function to be invertible, it must be both one-to-one (injective) and onto (surjective).

5 Must Know Facts For Your Next Test

1. A function $f$ is invertible if and only if it is bijective.
2. The graph of an invertible function will intersect any horizontal line at most once.
3. The inverse of a function $f$, denoted as $f^{-1}$, satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the domain.
4. If $f$ is a polynomial or rational function, we determine its invertibility by checking if it is strictly monotonic (always increasing or always decreasing).
5. The process of finding an inverse involves solving the equation $y = f(x)$ for $x$, then swapping the variables to get $x = f^{-1}(y)$.

Review Questions

• What conditions must a function satisfy to be considered invertible?
• How can you verify graphically if a given function is invertible?
• What are the steps involved in finding the inverse of a polynomial or rational function?

Related terms

Bijective Function: A bijective function is both injective (one-to-one) and surjective (onto), ensuring that every element in the codomain is mapped by exactly one element from the domain.

Monotonic Function: A monotonic function is one that is either entirely non-increasing or non-decreasing throughout its domain.

Horizontal Line Test: A graphical method used to determine if a function is injective; if no horizontal line intersects the graph more than once, the function passes this test.